The first official flag of the United States came in 1777 when the the 2nd Continental Congress passed The Flag Act of 1777, which, in its entirety, reads:

Resolved, That the flag of the thirteen United States be thirteen stripes, alternate red and white; that the union be thirteen stars, white in a blue field, representing a new constellation.

Since then, there have been two additional Flag Acts: one in 1794 and one in 1818. The latter act contained the following, crucial element:

And be it further enacted, That on the admission of every new state into the Union, one star be added to the union of the flag; and that such addition shall take effect of the fourth day of July then next succeeding such admission.

While the Flag Acts set out the number of stars and stripes in the flag, they didn't specify a particular arrangement for the stars. That would go unremedied until 1912, when President William Taft signed Executive Order 1556 - superseded by several other executive orders over the ensuing months - which noted the lack of a requirement for the proportions of the flag or for the arrangement of its stars, and declared an official arrangement for government use, which can be seen here.

After Hawaii and Alaska joined the Union, executive orders for new flags were made. The latter one, Executive Order 10834, from 1959, is still in force today, and is included in the US Government's publication of the US Flag Code, also known as Title 4, Chapter 1 of the United States Code.

So, in sum, Congress decided on the general parameters of the flag, and presidents, in a series of executive orders beginning in 1912, added specifics. In 1959, Dwight D. Eisenhower signed Executive Order 10834, which specified the standard we now know as the 50-starred US Flag.